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2.13: Second Derivatives and Laplace Operator

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Second Derivatives and Laplace Operator

2.13: Second Derivatives and Laplace Operator

The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.

Consider a scalar function. The curl of its gradient can be written as follows:


For a vector function, the divergence of a curl can be expressed as follows:


The curl of a gradient function and the divergence of a curl function are always zero.

The divergence of the gradient of a scalar function can be expressed as follows:


The Laplacian is analogous to the second-order differentiation of the scalar quantities. It describes physical phenomena like electric potentials and diffusion equations for heat flow.

The divergence of gradient and the curl of a curl are mathematical constructs. Lagrange's vector cross-product identity formula relates both to a vector Laplacian.


The vector Laplacian is obtained by directly applying the scalar Laplacian to each of the scalar components of a vector.


Suggested Reading


Second Derivatives Laplace Operator Del Operator Gradient Divergence Curl Second Order Expressions Mathematics Physics Scalar Function Vector Function Curl Of Gradient Divergence Of Curl Gradient Of Divergence Laplacian Electric Potentials Diffusion Equations Vector Laplacian Lagrange's Vector Cross-product Identity Formula

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